# Rose (mathematics)

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In mathematics, a **rose** or **rhodonea curve** is a sinusoid plotted in polar coordinates.

## Contents

## General overview[edit]

Up to similarity, these curves can all be expressed by a polar equation of the form

^{[1]}

or, alternatively, as a pair of Cartesian parametric equations of the form

If *k* is an integer, the curve will be rose-shaped with

- 2
*k*petals if*k*is even, and *k*petals if*k*is odd.

Where *k* is even, the entire graph of the rose will be traced out exactly once when the value of theta, θ changes from 0 to 2π; when *k* is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for *k* even, and π for *k* odd.)

If *k* is a half-integer (*e.g.* 1/2, 3/2, 5/2), the curve will be rose-shaped with 4*k* petals. Example: n=7, d=2, *k*= n/d =3.5, as θ changes from 0 to 4π.

If *k* can be expressed as *n*±1/6, where *n* is a nonzero integer, the curve will be rose-shaped with 12*k* petals.

If *k* can be expressed as *n*/3, where *n* is an integer not divisible by 3, the curve will be rose-shaped with *n* petals if *n* is odd and 2*n* petals if *n* is even.

If *k* is rational, then the curve is closed and has finite length. If *k* is irrational, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set (i.e., it comes arbitrarily close to every point in the unit disk).

Since

for all , the curves given by the polar equations

- and

are identical except for a rotation of π/2*k* radians.

Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728.^{[2]}

## Area[edit]

A rose whose polar equation is of the form

where *k* is a positive integer, has area

if *k* is even, and

if *k* is odd.

The same applies to roses with polar equations of the form

since the graphs of these are just rigid rotations of the roses defined using the cosine.

## How the parameter *k* affects shapes[edit]

In the form *k* = *n*, for integer n, the shape will appear similar to a flower. If *n* is odd half of these will overlap, forming a flower with *n* petals. However, if it is even the petals will not overlap, forming a flower with 2*n* petals.

When *d* is a prime number then *n*/*d* is a least common form and the petals will stretch around to overlap other petals; the number of petals each one overlaps is equal to the how far through the sequence of primes this prime is +1, i.e. 2 is 2, 3 is 3, 5 is 4, 7 is 5, etc.

In the form *k* = 1/*d* when *d* is even then it will appear as a series of *d*/2 loops that meet at 2 small loops at the center touching (0, 0) from the vertical and is symmetrical about the *x*-axis.
If *d* is odd then it will have *d* div 2 loops that meet at a small loop at the center from either the left (when in the form *d* = 4*n* − 1) or the right (*d* = 4*n* + 1).

If *d* is not prime and n is not 1, then it will appear as a series of interlocking loops.

If *k* is an irrational number (e.g. , , etc.) then the curve will have infinitely many petals, and it will be dense in the unit disc.

## Offset parameter[edit]

Adding an offset parameter *c*, so the polar equation becomes

alters the shape as illustrated at right. In the case where the parameter *k* is an odd integer, the two overlapping halves of the curve separate as the offset changes from zero.

## Programming[edit]

```
k <- 4
t <- seq(0, 4*pi, length.out=500)
x <- cos(k*t)*cos(t)
y <- cos(k*t)*sin(t)
plot(x,y, type="l", col="blue")
```

MATLAB and OCTAVE

```
function rose(del_theta, k, amplitude)
% inputs:
% del_theta = del_theta is the discrete step size for discretizing the continuous range of angles from 0 to 2*pi
% k = petal coefficient
% if k is odd then k is the number of petals
% if k is even then k is half the number of petals
% amplitude = length of each petal
% outputs:
% a 2D plot from calling this function illustrates an example of trigonometry and 2D Cartesian plotting
theta = 0:del_theta:2*pi;
x = amplitude*cos(k*theta).*cos(theta);
y = amplitude*cos(k*theta).*sin(theta);
plot(x,y)
```

## See also[edit]

- Lissajous curve
- quadrifolium – a rose curve where
*k*= 2. - Maurer rose
- Rose (topology)
- Spirograph

## Notes[edit]

**^***Mathematical Models*by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 73.**^**O'Connor, John J.; Robertson, Edmund F., "Rhodonea",*MacTutor History of Mathematics archive*, University of St Andrews.

## External links[edit]

Wikimedia Commons has media related to .Rose curves |